The Basis Number of the Cartesian Product of Different Ladders

Abstract

The basis number of a graph $G$ is defined to be the least positive integer $d$ such that $G$ has a $d$-fold basis for the cycle space of $G$.

In this paper, we prove that the basis number of the Cartesian product of different ladders is exactly $4$. However, if we apply Theorem $4.1$ of Ali and Marougi $[4]$, which is stated in the introduction as Theorem $1.1$, we find that the basis number of the circular and Möbius ladders with circular ladders and Möbius ladders is less than or equal to $5$, and the basis number of ladders with circular ladders and circular ladders with circular ladders is at most $4$.